The true discount on a certain sum of money due 3 years hence is Rs. 100 and the simple interest on the same sum for the same time and at the same rate is Rs. 120. What is the rate of interest per annum?

Introduction / Context:
This question compares true discount and simple interest for the same sum, same period, and same rate. The true discount and simple interest values are given. The target is the rate of interest. This is a standard pattern in aptitude exams and requires understanding how true discount is related to simple interest through the effective rate time product r * t.

Given Data / Assumptions:

• True discount TD = Rs. 100.
• Simple interest SI = Rs. 120.
• Time t = 3 years.
• Rate of interest r percent per annum is unknown.
• Both TD and SI are calculated on the same sum S and at the same rate r for the same time t.

Concept / Approach:
For a sum S due after time t at rate r percent, we have: SI = S * r * t / 100 TD = S * r * t / (100 + r * t) If SI and TD for the same sum and time are known, then the ratio SI / TD is: SI / TD = (S * r * t / 100) / (S * r * t / (100 + r * t)) = (100 + r * t) / 100. Thus: SI / TD = (100 + r * t) / 100. We use the given numerical values to find r * t and then r.

Step-by-Step Solution:
Step 1: Compute SI / TD from the given data. SI / TD = 120 / 100 = 6 / 5. Step 2: Set this equal to (100 + r * t) / 100. 6 / 5 = (100 + r * t) / 100. Step 3: Cross multiply: 6 * 100 = 5 * (100 + r * t). 600 = 500 + 5 * r * t. Step 4: 5 * r * t = 600 − 500 = 100. Step 5: So r * t = 100 / 5 = 20. Step 6: Time t is 3 years, so r = (r * t) / t = 20 / 3. Step 7: 20 / 3 percent is 6 whole and 2 thirds percent, that is (6 + 2/3)%.

Verification / Alternative check:
We can find the sum S and verify. Using SI = S * r * t / 100: 120 = S * 20 / 100, so S = 120 * 100 / 20 = 600. Now compute TD with S = 600 and r * t = 20: TD = S * r * t / (100 + r * t) = 600 * 20 / 120 = 600 * 1 / 6 = 100. This equals the given true discount, confirming that r = 20 / 3 percent, or (6 + 2/3) percent, is correct.

Why Other Options Are Wrong:
Rates like (5 + 2/3) percent, (4 + 2/3) percent, or (2 + 2/3) percent correspond to smaller values of r * t and would not satisfy SI / TD = 6 / 5. Substituting them into the relationship leads to a different ratio between SI and TD, which contradicts the given data.

Common Pitfalls:
One frequent mistake is to treat SI and TD as equal or to ignore the denominator 100 + r * t in the true discount formula, which destroys the key relationship. Others incorrectly cancel variables without considering ratios. Using the ratio SI / TD is a clean and systematic way to avoid dealing with the unknown sum S directly and leads quickly to the effective rate time product and then the rate itself.

Final Answer:
The annual rate of interest is (6 + 2/3)%.